spin dynamics of the quasi-two-dimensional spin-1 quantum ... · spin dynamics of the...

Spin dynamics of the quasi-two-dimensional spin-12 quantum magnet Cs2CuCl4

M. Y. Veillette, A. J. A. James, and F. H. L. EsslerRudolf Peierls Centre for Theoretical Physics, University of Oxford, 1, Keble Road, Oxford, OX1 3NP, United Kingdom

�Received 24 June 2005; revised manuscript received 24 August 2005; published 31 October 2005�

We study dynamical properties of the anisotropic triangular quantum antiferromagnet Cs2CuCl4. Inelasticneutron scattering measurements have established that the dynamical spin correlations cannot be understoodwithin a linear spin wave analysis. We go beyond linear spin wave theory by taking interactions betweenmagnons into account in a 1/S expansion. We determine the dynamical structure factor and carry out extensivecomparisons with experimental data. We find that compared to linear spin wave theory, a significant fraction ofthe scattering intensity is shifted to higher energies and strong scattering continua are present. However, the1/S expansion fails to account for the experimentally observed large quantum renormalization of the exchangeenergies.

DOI: 10.1103/PhysRevB.72.134429 PACS number�s�: 75.10.Jm, 75.25.�z, 75.30.Ds, 75.40.Gb

I. INTRODUCTION

The quasi-two-dimensional spin-12 quantum magnet

Cs2CuCl4 has attracted much theoretical and experimentalinterest in recent years as a possible realization of a two-dimensional quantum spin liquid.1–13 This anisotropic trian-gular Heisenberg antiferromagnet is believed to be a prom-ising candidate due to its small spin, quasi-two-dimensionality and geometrically frustrated spin interactions.Although Cs2CuCl4 exhibits conventional incommensuratelong range magnetic order at low temperatures, neutron scat-tering measurements have revealed unusual features in thespin excitation spectrum. In particular, the dynamical corre-lations are found to be dominated by an extended scatteringcontinuum over a relatively large window of energies. Sev-eral workers have interpreted this observation as a signatureof deconfined, fractionalized spin-1

2 �spinon� excitations,characteristic of a spin liquid phase. In this line of approach,the observed broad scattering continuum is interpreted interms of a two-spinon scattering continuum.7,9,13

However, a strong scattering continuum does not entail anunderlying spin liquid phase. In fact, a conventional mag-netically ordered phase with strong magnon interactions canexhibit a broad continuum due to multiple magnon scatteringprocesses. A previous examination of the inelastic neutronscattering data on Cs2CuCl4 was performed in the frameworkof linear spin wave �LSW� theory.4 The latter predicts sharpsingle-particle excitations and weak two-magnon scatteringcontinua: features which were argued to be in poor agree-ment with the data. Given that the magnetic properties derivefrom small S= 1

2 Cu spins, one would a priori expect magnoninteractions to play an important role. In order to assess theapplicability of a spin wave based scenario to Cs2CuCl4 it is,therefore, necessary to go beyond linear spin wave theory.

On a qualitative level, the predictions of nonlinear spinwave theory are readily anticipated. By Goldstone’s theorem,the breaking of a continuous symmetry in a magneticallyordered state enforces the presence of single-particle excita-tions at low energies. As a result of the aforementioned in-teractions, these magnons acquire a finite lifetime, which inturn leads to a finite linewidth in the dynamical structure

factor. Furthermore, compared to linear spin wave theory,spectral weight is transferred to higher energies via multiplemagnon scattering processes. In the case of Cs2CuCl4 onemay expect the presence of a strong scattering continuum inthe ordered phase because �i� the low spin and the frustratednature of the exchange interactions lead to a small orderedmoment and strong quantum fluctuations around the orderedstate; �ii� the magnon interactions in noncollinear spin struc-tures like the ones found in Cs2CuCl4 induce a couplingbetween transverse and longitudinal spin fluctuations. Thisinteraction provides an additional mechanism for dampingthe spin waves and can enhance the strength of the scatteringcontinuum.

There is evidence of low-energy spin wave modes in theinelastic neutron scattering data. Sharp peaks are also ob-served at high energies near special wave vectors where aputative spin wave dispersion is at a saddle point. It is im-portant to note that this spin wave dispersion is dramatically“renormalized” compared to the prediction of linear spinwave theory.3,4

A priori it appears that nonlinear spin wave theory couldhave the necessary ingredients to account for the spin corre-lations observed in Cs2CuCl4. The issue then is whether it ispossible to achieve a quantitative description of the experi-ments in low orders of perturbation theory in the spin waveinteractions.

In the present work, we go beyond linear spin wavetheory and include, within the framework of a 1/S expan-sion, the quantum fluctuations around the classical groundstate. We then apply the results to the case S= 1

2 , in whichthe formal expansion parameter becomes of order 1 and,therefore, is not small. We are motivated by the observationthat spin wave theory gives a good description of physicalproperties of the square-lattice spin-1

2 HeisenbergHamiltonian.14–16 Indeed, higher-order �in a 1/S expansion�corrections to linear spin wave theory were shown to besmall in this case. Furthermore, taking these corrections intoaccount in the calculation of static and dynamical propertiesleads to an improved agreement with the results of moresophisticated numerical techniques.17–19 Although a corre-sponding analysis is not available for the frustrated triangular

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antiferromagnet, perturbative expansions in 1/S have shownthe renormalization due to quantum effects is relativelysmall.20–23

This paper is organized as follows. The spin Hamiltonianfor Cs2CuCl4 is introduced in Sec. II. In Sec. III, we deter-mine the magnon Green’s function in the framework of alarge-S expansion. In Sec. IV, we relate the experimentallymeasured dynamical correlation functions to the magnonGreen’s function. The results of our analysis and compari-sons to the experimental data on Cs2CuCl4 are presented inSec. V. We conclude with a summary of our results in Sec.VI.

II. SPIN MODEL

The full spin Hamiltonian of Cs2CuCl4 has been deter-mined previously from measurements in high magnetic fields�see Ref. 3 for details�. For our purposes, it suffices to notethat the magnetic Cu2+ ions form a triangular lattice withanisotropic exchange interactions. As shown in Fig. 1, themain exchange interaction J=0.374�5� meV is along thecrystallographic b axis �“chain direction”�. A weaker spinexchange J�=0.128�5� meV occurs along the zig-zag bonds.Finally, a Dzyaloshinskii-Moriya �DM� interaction24,25 D=0.020�2� meV is present along the zigzag bonds.

Denoting the spin-12 operators at the sites R by SR, the

quasi-two-dimensional Hamiltonian takes the form

H = �R

JSR · SR+�1+�2+ J��SR · SR+�1

+ SR · SR+�2�

− �− 1�nD · SR � �SR+�1+ SR+�2

� . �1�

Here the vectors �1 and �2 connecting neighboring sites areshown in Fig. 1. The vector D= �D ,0 ,0� is associated withthe oriented bond between the two coupled spins connectedby �1 or �2 and n is a layer index. The factor �−1�n indicatesthat the interaction alternates between even and odd layers,which as a result can be considered to be inverted versions ofone another. A weak interlayer interaction J� is also presentbetween neighboring layers. However, as J� is quite small,we neglect it in the following.

Following the conventions of Coldea et al. in Ref. 4, wewill discuss the dynamic response in terms of the two-

dimensional Brillouin zone of the triangular lattice eventhough the full crystal symmetry is orthorhombic �see Fig.2�. In our notation, wave vectors are expressed in terms ofthe reciprocal lattice vectors as k= �h ,k , l�, which is a short-hand for 2��h /a ,k /b , l /c�.

The Fourier transforms of the exchange and DM interac-tions are

JQ = J cos�2�k� + 2J� cos��k�cos��l� �2�

and

DQ = − 2iD sin��k�cos��l� . �3�

It is convenient for what follows to define a quantity

JQT = JQ − iDQ. �4�

Experimentally, spiral magnetic long range order is observedin Cs2CuCl4 at temperatures below TN=0.62�1� K. The or-dered structure is found to lie in the bc plane by virtue of thesmall easy-plane anisotropy generated by the DM interac-tions. The spin structure is an incommensurate cycloid withan ordering wave vector Q= �0.0,0.5+� ,0�, where �=0.030�2�.

III. LARGE-S EXPANSION

We now turn to a summary of our calculations. The pro-cedure we follow is standard. We first express thefluctuations around the “classical” ground state in terms ofboson operators using the Holstein-Primakofftransformation.21–23,26–30 The term quadratic in the boson op-erators constitutes the basis for linear spin wave theory,whereas higher-order terms represent spin wave interactions.The interaction vertices of n bosons carry a factor S2−n/2,where S is the “length” of the spin. In the second step, wedetermine the renormalized magnon Green’s function by cal-culating the self-energy to the leading order in 1/S. Finally,the experimentally observable dynamical correlation func-tions are expressed in terms of the Green’s function of theHolstein-Primakoff bosons.

The classical ground state is determined by treating thespins as classical vectors and then minimizing the energy. In

FIG. 1. �Color online� The magnetic sites and exchange cou-plings within a single layer of Cs2CuCl4. Layers are stacked alongthe crystallographic a direction with an interlayer spacing a /2 and arelative displacement in the c direction.

FIG. 2. �Color online� The reciprocal space diagram ofCs2CuCl4 projected along the �0,k , l� plane. The � points refer tothe center of the Brillouin zone and Q is the ordering wave vector.The path of the cut shown in Fig. 5 is depicted as a dashed line.

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this way, one obtains a cycloidal structure with a character-istic wave vector Q that is fixed by the condition that itminimizes the exchange energy per spin, i.e., JQ

T =minq JqT.

We find Q= �0.0,0.5+�0 ,0� with �0=0.054. This value dif-fers significantly from the measured incommensuration butquantum fluctuations lead to a reduction in �0 and takingthem into account yields good agreement withexperiments.7,31

As we have already indicated in Eq. �1�, to a good ap-proximation the layers are decoupled. Hence, we considerfrom now on a set of independent two-dimensional �2D� lay-ers, which are subdivided into two groups, differing accord-ing to the direction of the DM vector. For the case where thelayer index n is odd �even�, the DM vector is taken to pointinto �out of� the bc plane.

In what follows, we present the results for the even layersonly. However, it is easy to see that the spin structure factoris, in fact, independent of the layer index and the overallresult is a simple summation over all layers.

It is convenient to define a local reference frame �x ,y ,z�such that the classical spin direction is aligned along the zaxis at every site

�SRa

SRb

SRc � = �1 0 0

0 cos�Q · R� − sin�Q · R�0 sin�Q · R� cos�Q · R�

��SRx

SRy

SRz � . �5�

The Holstein-Primakoff transformation reads26

SR+ = SR

x + iSRy = ei��2S − �R

† �R��R,

SR− = SR

x − iSRy = e−i��R

† ��2S − �R† �R� ,

SRz = S − �R

† �R, �6�

where the boson creation and annihilation operators satisfythe canonical commutation relation ��R ,�R�

† �=�R,R�. Here �

is an arbitrary angle which we set equal to � /2 in order tomake contact with the notation used in Ref. 22. Introducingthe Fourier transform,

�k† =

1�N

�R

�R† e−ik·R, �7�

on a lattice of N sites, the Hamiltonian of Eq. �1� takes theform

H = H0 + H2 + H3 + H4 + ¯ , �8�

where Hn is proportional to S2−n/2 and consists of normalordered products of n boson operators. There is no H1 term,because Eq. �8� is an expansion around a minimum of theclassical energy. Linear spin wave theory takes into accountonly the terms H0 and H2. The higher-order terms representinteractions between magnons. The leading terms in the ex-pansion are

H0 = NS2JQT , �9�

H2 = NSJQT + S�

kAk��k

†�k + �−k�−k† � − Bk��−k

† �k† + �−k�k� ,

�10�

H3 =i

2� S

2N�

1,2,3�1+2+3�C1 + C2��−3

† �2�1 − �1†�2

†�−3� ,

�11�

H4 =1

4N�

1,2,3,4�1+2+3+42

3�B2 + B3 + B4��1

†�−2�−3�−4

+ �−4† �−3

† �−2† �1� + ��A1+3 + A1+4 + A2+3 + A2+4� − �B1+3

+ B1+4 + B2+3 + B2+4� − �A1 + A2 + A3

+ A4��1†�2

†�−3�−4 . �12�

Here the sum over k is performed in the first Brillouin zoneand the subscripts 1 … 4 denote k1…k4. The quantities Ak,Bk, and Ck are expressed as

Ak =1

4�2Jk + JQ+k

T + JQ−kT � − JQ

T ,

Bk =1

4�2Jk − JQ+k

T − JQ−kT � ,

Ck = JQ+kT − JQ−k

T . �13�

The coefficients Ak and Bk are even functions of k, whereasCk is an odd function of k. In the absence of easy-planeanisotropies, i.e., when D vanishes and inversion symmetryis present, we recover the results of Ref. 22. �Note that ourdefinitions in Eqs. �13� differ from those of Ref. 22 by afactor of 4.� We emphasize that the cubic interaction is gen-erated as a result of the coupling between transverse andlongitudinal fluctuations and, hence, can only exist in non-collinear spin structures. Furthermore, we note that the ver-tex factor Ck �k�3 for small k owing to the fact that JQ

T is ata minimum by construction.

The quadratic Hamiltonian H2 is diagonalized by a Bogo-liubov transformation

�k = ukk + vk−k† ,

�−k† = vkk + uk−k

† , �14�

where

uk2 = 1 + vk

2 =1

2� Ak

�Ak2 − Bk

2+ 1 ,

ukvk =1

2

Bk

�Ak2 − Bk

2. �15�

The diagonal form of the quadratic Hamiltonian is

SPIN DYNAMICS OF THE QUASI-TWO-DIMENSIONAL … PHYSICAL REVIEW B 72, 134429 �2005�

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H2 = NSJQT + �

k�k�k

†k +1

2 , �16�

where �k=2S�Ak2 −Bk

2 is the linear spin wave dispersionrelation.27,28 We note that �k is an even function of k, despitethe absence of inversion symmetry in the Hamiltonian. Infact, the symmetry of �k is a consequence of time-reversalsymmetry, which implies the following relation between theelements of the dynamical structure factor �Eq. �27��,32

Sk,�� = S−k,�

� . �17�

The importance of quantum fluctuations can be gauged bydetermining the average value of the local spin given by thestandard formula

�SRz � = S − �S = S −

1

2N�k

uk2 + vk

2 . �18�

The boson Green’s function at zero temperature is expressedas

Gk,� = − i�−�

�

dtei�t�T� �k�t��−k

† �t� ��k†�0��−k�0�� , �19�

where T denotes time ordering and �…� represents a groundstate expectation value. The inverse of the unperturbedGreen’s function is given by a 2�2 matrix,

Gk,��0�−1 = �− 2SAk + i��0 + 2SBk�x + ��z. �20�

Here �0 and � denote the identity and Pauli matrices, respec-tively, and �=0+.

The self-energy is defined by the Dyson equation,

Gk,�−1 = Gk,�

�0�−1 − �k,�, �21�

and can be parametrized as

�k,� = Ok,��0 + Xk,��x + Zk,��z. �22�

The leading order �in 1 /S� contributions to the self-energycan be divided into two parts

�k,� = �k�4� + �k,�

�3� . �23�

Here �k�4� denotes the vacuum polarization contribution

that arises in first-order perturbation theory in H4. It is fre-quency independent and purely real. On the other hand, �k,�

�3�

denotes the contribution in second-order perturbation theoryof the three-magnon interaction H3. It incorporates the ef-fects of magnon decay. Using Eq. �20�, the �k

�4� contributionto the self-energy is found to be of the form

Ok�4� = Ak +

2S

N�k�

1

�k��1

2Bk + Bk� Bk�

+ �Ak−k� − Bk−k� − Ak� − Ak�Ak�� ,

Xk�4� = − Bk +

2S

N�k�

1

�k��Bk + Bk��Ak�

+ �Ak−k� − Bk−k� − Ak� −1

2Ak Bk�� ,

Zk�4� = 0. �24�

The contribution ��3� is most easily evaluated in the Bo-goliubov basis �� and is equal to

Ok,��3� =

− S

16N�k�

��1��k�,k − k��2 + ��2��k�,k − k��2�

�� 1

�k� + �k−k� − � − i�+

1

�k� + �k−k� + � − i� ,

Xk,��3� =

− S

16N�k�

��1��k�,k − k��2 − ��2��k�,k − k��2�

�� 1

�k� + �k−k� − � − i�+

1

�k� + �k−k� + � − i� ,

Zk,��3� =

− S

16N�k�

�2��1��k�,k − k��2��k�,k − k��

�� 1

�k� + �k−k� − � − i�−

1

�k� + �k−k� + � − i� ,

�25�

where

��1��k�,k − k�� = �Ck� + Ck−k��uk� + vk��uk−k� + vk−k��

− 2Ck�uk�vk−k� + vk�uk−k�� ,

��2��k�,k − k�� = Ck��uk� + vk��uk−k� − vk−k�� + Ck−k��uk−k�

+ vk−k��uk� − vk�� . �26�

IV. DYNAMICAL CORRELATION FUNCTION

Inelastic neutron scattering experiments probe the dy-namical structure factor Sk,�

� . The latter is defined as the Fou-rier transform of the dynamical spin-spin correlation function

Sk,�� = �

−�

� dt

2�e−i�t�S−k

� �0�Sk �t�� . �27�

Here � , = �a ,b ,c� label the various crystallographic axesand the Fourier-transformed spin operators are defined bySk

�= 1�N

�RSR�e−ik·R.

It is convenient to introduce time-ordered spin-spin corre-lation functions in the rotated coordinate system

Fk,�� = − i�

−�

�

dte−i�t�TS−k� �0�Sk

��t�� , �28�

where � ,�= �x ,y ,z� are the rotated coordinate axes �Eq. �5�,see Fig. 3�. The dynamical structure factor is related to the


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imaginary part of the time-ordered correlation function in thefollowing way:

Sk,�aa = −

1

�Im Fk,�

xx , �29�

Sk,�bb = Sk,�

cc = −1

�Im��k+Q,�

+ + �k−Q,�− � , �30�

Sk,�bc = − Sk,�

cb = −i

�Im��k+Q,�

+ − �k−Q,�− � , �31�

where

�k,�± =

1

4�Fk,�

zz + Fk,�yy ± i�Fk,�

zy − Fk,�yz �� . �32�

To proceed further, we expand the dynamical correlationfunctions in inverse powers of S to order O�S0�. The corre-sponding results have been derived previously by Ohyamaand Shiba.22 Here we merely quote their results for the sakeof completeness. The transverse correlations are

Fk,�xx =

S

2cx

2 Tr��0 − �x�Gk,�� ,

Fk,�yy =

S

2cy

2 Tr��0 + �x�Gk,�� , �33�

where the Green’s function is given by Eq. �21� and where

cx = 1 −1

4SN�k

�2vk2 − ukvk� ,

cy = 1 −1

4SN�k

�2vk2 + ukvk� . �34�

We note that when squaring �34� only terms to order O�S−1�must be retained. The mixing of transverse and longitudinalfluctuations manifests itself in

i�Fk,�yz − Fk,�

zy � = cy�Pk,��1� Tr��0 + �x�Gk,�� + Pk,�

�2� Tr��zGk,�� .

�35�

Here the functions Pk,��1,2� are defined as

Pk,��1� =

S

4N�k�

��1��k�,k − k��uk�vk−k� + vk�uk−k��

�� 1

�k� + �k−k� − � − i�+

1

�k� + �k−k� + � − i� ,

Pk,��2� =

S

4N�k�

��2��k�,k − k��uk�vk−k� + vk�uk−k��

�� 1

�k� + �k−k� − � − i�−

1

�k� + �k−k� + � − i� .

�36�

Finally, the longitudinal correlations are decomposed in in-verse powers of S as Fk,�

zz =Fk,��0�zz+Fk,�

�1�zz, where

Fk,��0�zz = −

1

2N�k�

�uk�vk−k� + vk�uk−k��2� 1

�k� + �k−k� − � − i�

+1

�k� + �k−k� + � − i� , �37�

Fk,��1�zz =

1

2S��Pk,�

�1� �2

�Tr��0 + �x�Gk,�� + �Pk,��2� �2Tr��0 − �x�Gk,��

+ 2Pk,��1� Pk,�

�2� Tr��zGk,�� . �38�

We note that the F�0�zz term does not require the knowl-edge of the bosonic self-energy and is basically a free bosonresult. For this reason, it is often included in linear spin wavecalculations as a source of two-magnon scattering, eventhough it is formally a higher-order contribution in 1/S. Inwhat follows, we abide by this �in some sense inconsistent�convention and consider the contribution of Eq. �37� as partof the linear spin wave theory. As a consequence, we thenretain the F�1�zz contribution to the dynamical structure fac-tor, although it is of higher order in 1/S �i.e., O�S−1�� thanthe other terms we take into account.

The �unpolarized� inelastic neutron scattering cross sec-tion is given by

d2�

d�d�= �fk�2�

�

�� − k̂�k̂ �Sk,�� ,

= �fk�2��1 − k̂a2�Sk,�

aa + �1 + k̂a2�Sk,�

bb � , �39�

where k̂� is the � component of the unit vector in k direc-tion. The magnetic form factor fk is determined by the mag-netic ions. For Cu2+, the isotropic form factor has a relativelyweak wave vector dependence within the first Brillouin zoneand will be neglected from now on.33

It is well known that the 1/S expansion preserves manyphysical properties “order by order” in 1/S. For instance, it

FIG. 3. �Color online� Incommensurate spiral order. The thickarrows represent the spin direction of the spiral order. The crystal-lographic axes �a ,b ,c� and the local reference frame of the spins�x ,y ,z� are indicated. The spins lie in the bc plane and the anglebetween consecutive spins along the b direction is �=2��0.5+�0�=199°. In the rotating reference frame, the z axis is the average spindirection, whereas the x and y axes are the transverse directions.


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follows from Eqs. �24� and �25� that the Goldstone modespersist beyond linear spin wave theory, as one expects, onphysical grounds. A careful examination also shows that toorder O�S0� the spectral functions are positive and that therelation �17� holds. However, due to a lack of self-consistency, the 1/S expansion leads to an �unphysical� un-equal treatment of the one-magnon and two-magnon scatter-ing contributions to dynamical correlation functions.23 It isworthwhile to discuss this issue in more detail. The leadingorder contribution to the dynamical structure factor is due tocoherent single-magnon excitations and is of the form ��−�k�. The two-magnon contribution due to longitudinal fluc-tuations �Eq. �37�� gives rise to a scattering continuum of theform �k�I�k ,k��−�k�−�k−k�� with some functionI�k ,k��. The extent of the two-magnon contribution in k−� space is determined by the lower and upper bounds ofthe function �k�+�k−k� for a given k.

On general grounds, we expect the lower bound of thetwo-magnon scattering continuum to be equal to or smallerthan the “true” magnon dispersion �̄k. In fact, the existenceof a zero-momentum Goldstone mode guarantees that thereexists a two-magnon contribution at frequencies �̄k+ �̄0= �̄k.

It is easy to see that this property does not hold order byorder in a 1/S expansion. Indeed, the first-order contributionin 1/S shifts the pole of the Green’s function and leads to arenormalization of the magnon dispersion. The renormalizeddispersion �̃k can be determined from the Dyson equation,

Gk,�̃k

−1 = Gk,�̃k

�0�−1 − �k,�̃k= 0. �40�

However, to order O�S0�, the threshold of the two-magnoncontribution is still determined by the bare dispersion rela-tion �k. This results in an unphysical behavior, where thetwo-magnon scattering continuum is separated from thesingle-magnon dispersion by a gap. In order to avoid thisproblem, we impose the following self-consistency condi-tion: the linear spin wave dispersion �k used in Eqs. �25�,�36�, and �37� is to be replaced by the renormalized disper-sion �̃k.

V. DYNAMICAL PROPERTIES OF Cs2CuCl4

So far our discussion of the 1/S expansion has been fairlygeneral. In order to make contact with the experiments onCs2CuCl4, we now set the exchange constants to their appro-priate values3,4 and fix S= 1

2 . We then evaluate the dynamicalstructure factor at a given wave vector numerically. Complexintegrals such as Eqs. �25� and �36� are evaluated by sum-ming the imaginary part of the integrands over a frequencygrid of 1200 points and of 1000�1000 points in wave vectorspace. The real parts are then determined from the Kramers-Kronig relations. The aforementioned self-consistency con-dition is implemented by calculating the full Green’s func-tion iteratively on a 100�100 grid in the Brillouin zone. Weobserve satisfactory numerical convergence after about threeiterations.

We first turn to the magnon dispersion. The linear spinwave result �k vanishes at the center of the paramagnetic

Brillouin zone. The corresponding Goldstone mode is asso-ciated with small fluctuations of the ordered moment withinthe cycloidal plane. In helimagnets, the spectrum often ex-hibits a second Goldstone mode at the ordering wave vector.This gapless mode is due to fluctuations of the plane of thecycloid. In the case at hand, the easy-plane anisotropy gen-erated by the DM term forces the cycloidal structure to lie inthe bc plane and creates an excitation gap at the orderingwave vector Q.

The renormalization of the magnon dispersion within theframework of the 1/S expansion is obtained from the polesof the Green’s function �Eq. �40��. In Fig. 4, we compare theresults of the 1/S expansion with the linear spin wave theory.It is customary to quantify the effects of the “quantum”renormalization of the magnon dispersion by parametrizing

the latter in terms of “effective” exchange constants J̃, J̃�,

and D̃ and comparing them with the “bare” parameters J, J�,and D.

Experimentally, the quantum renormalization is found to

be rather large, namely J̃ /J=1.63�5� and J̃� /J�=0.84�9�. Therenormalization of D was not established. The 1/S expansion

yields the significantly smaller renormalizations J̃ /J=1.131,

J̃� /J�=0.648, and D̃ /D=0.72. The difference between thetheoretical and experimental values indicates that the leadingorder in a 1/S expansion underestimates fluctuation effects.On the other hand, one should note that the 1/S expansiongives a result of 0.031 for the incommensuration, which isvery close to the experimentally observed value of �=0.030�2�.

Before turning to a comparison of our results for the dy-namical structure factor with the experimental results, webriefly review some facts about excitations in helimagnets.Generally it is useful to distinguish between three spin wavemodes. In the case at hand, the “principal” mode �k

0 = �̃k ispolarized along the a axis and is probed by the Sk,�

aa compo-nent of the dynamical structure factor �Eq. �29��. The two“secondary” spin wave modes �k

±= �̃k±Q are images of theprincipal mode but their momenta are shifted by ±Q. They

FIG. 4. �Color online� The renormalization of the spin wavespectrum. The solid and dashed lines are results of the 1/S expan-sion �̃k �Eq. �40�� and the linear spin wave dispersion �k �Eq. �16��.The cut in the paramagnetic Brillouin zone runs from �000� to �010�to �011� and back to the center of the Brillouin zone �000�.


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are polarized in the bc plane �Eqs. �30� and �31��. In linearspin wave theory, the three spin wave modes give rise tosharp � functions and carry a large part of the spectralweight.

In addition to the single-magnon modes, there are mul-tiple magnon scattering continua. Whenever the magnon dis-persion lies within a scattering continuum, the single-magnon excitation gets broadened and acquires a finitelinewidth. On the other hand, when the magnon dispersionlies at the threshold of a scattering continuum, there is nosignificant decay and the single-magnon mode remainssharp.

The unpolarized dynamical structure factor �where thevarious modes are added according to Eq. �39�� is shown inFig. 5 as a function of energy and momentum for a particular“cut” of momentum transfers. The cut along the b direction,i.e., from �000� to �010�, shows large modulations of thedispersion relation due to the strong intrachain correlations.Near the ordering wave vector Q, the scattering intensityincreases sharply. For momentum transfers perpendicular tothe chains �i.e., along the �01�� direction�, the single-particlemodes are seen to be resolution limited. The two in-planemodes become degenerate and their dispersions are nearlyfeatureless, whereas the out-of-plane fluctuations dip to zeroenergy at �011�, in accordance with Goldstone’s theorem.Along the �0�� direction, the spectrum is symmetric acrossthe Brillouin zone boundary. Additional structures due totwo-magnon scattering are clearly visible at higher energiesalong the �0�0� and �0�� directions.

In order to illustrate how the spectral weights associatedwith the single-particle excitations are affected by the mag-non interactions, we have estimated their contributions foreach mode to the integrated spectral weights. The total inte-

grated intensity of each mode is given by “equal-time” cor-relation functions,

Ik0 = −

1

�� d�Fk,�

xx ,

Ik± = −

1

�� d��k±Q,�

± . �41�

The one-magnon contribution to the integrated intensity ofeach mode is then determined by integrating the respectivecorrelation function in the vicinity of the single-particle dis-persions. In practice, we find that integrating the peaks as-suming a Lorentzian form is a poor prescription for stronglydamped peaks. Instead, we numerically integrate the inten-sity over an energy window of three times the width at halfmaximum

Rk0 =

1

Ik0�

�k0−1.5��k

0

�k0+1.5��k

0

d�− Fk,�

xx

�,

Rk± =

1

Ik±�

�k±Q± −1.5��k±Q

±

�k±Q± +1.5��k±Q

±

d�− �k±Q,�

±

�. �42�

The results are shown in Fig. 6. We see that the integratedspectral weight is concentrated in the vicinities of the order-ing wave vector Q and �0 1

212

� and is largely suppressed nearthe � point. The weights associated with single-magnon ex-citations are strongly suppressed for the secondary modes.This is a consequence of the noncollinearity of the magneticorder. The in-plane modes are significantly damped as a re-sult of the coupling between longitudinal and transverse fluc-tuations. Such a coupling is not present for the out-of-plane

FIG. 5. �Color online� Density plot of the scattering cross section �Eq. �39�� as a function of energy and wave vector. The light �dark�regions represent regions of small �large� scattering intensity. The results have been convolved with the experimental energy resolution of thedetectors �the full width at half maximum is �E=0.016 meV�. The magnetic form factor of copper in Eq. �39� shows very weak wave vectordependence in the regime of interest and, therefore, was taken to be at unity �Ref. 33�. The filled circles along the �010� direction are theexperimental position of the most intense peaks in the line shapes taken in the spiral phase �T�0.1 K� �Ref. 4�.


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mode and, therefore, the principal mode carries generallymore spectral weight. Nevertheless, the fraction of spectralweight associated with single-particle excitations decreasessignificantly whenever the renormalized spin wave disper-sion is pushed upward in energy for a given momentum. Forinstance, near wave vector �0,0.8,0�, the principal spin wavemode lies within the two-magnon continuum and, as a result,less than 50% of the spectral weight is attributed to the one-magnon excitation.

The scattering intensity can also be studied by performinga wave vector average,

IT�� =1

N�k

��

Sk,�� . �43�

By the frequency sum rule, the scattering intensity �Eq. �43��integrated over all energies �including the elastic Braggpeaks at �=0� has to equal S�S+1�. However, this sum ruledoes not hold order by order in perturbation theory. For in-stance, the total intensity within linear spin wave theory ex-ceeds the sum rule by �S�1+2�S�. Bearing this caveat inmind, the sum rule is a useful tool for comparing the one-magnon and two-magnon contributions as well as analyzingthe shift in spectral weight. In Fig. 7, we plot the scatteringintensities as functions of energy within linear spin wave

theory and the 1/S expansion. In linear spin wave theory, theintegrated intensity exhibits cusps, which are associated withvan Hove singularities in the single-particle density of states.In the 1/S expansion such sharp features are absent. Aboveapproximately 0.5 meV, the one-magnon contribution van-ishes and the scattering intensity is entirely due to multiplemagnon states.

To quantify the shift of the spectral weight, we calculatethe first moment of the normalized scattering intensity ��.We find that the linear spin wave theory value ��=0.35 meV is renormalized upward to ��=0.40 meV in the1/S expansion. This observation is in line with the expecta-tion that the higher orders of the 1/S expansion induce atransfer of spectral weight to higher energies via multiplemagnon scattering processes. In fact, as shown in Fig. 8, thetwo-magnon contribution to the overall intensity is 29% inlinear spin wave theory, but 46% in the 1/S expansion.

Excitation line shapes

In order to exhibit the properties of the dynamical struc-ture factor in greater detail, we have generated a series ofscans in k−� space. The inelastic neutron scattering mea-surements on Cs2CuCl4 were not performed at constant mo-

FIG. 6. �Color online� Upper panel �a�: Theintegrated spectral weights for the three modes�Eq. �41�� as functions of momentum transfer.The principal �0 and secondary �+ ,�− spin wavemodes are defined in the text. Lower panel �b�:The ratios of the spectral weights of the single-particle excitations of each mode to their respec-tive integrated intensities �Eq. �42��.

FIG. 7. �Color online� The scattering intensityas a function of energy �Eq. �43�� for LSW theory�dashed line� and LSW+1/S expansion �solidline�. The contributions of the scattering continuaare shown using thin lines.


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mentum transfer, but followed various trajectories in energy-wave-vector space. We have generated our theoretical scansusing the known parameterizations of the scans A to J of Ref.4 in k−� space, which we summarize in Table. I. We referthe reader to Ref. 4 for further details. The various scans areshown in Fig. 9. Also shown are the regions in which sig-nificant magnetic scattering is observed experimentally and

the location of the main peaks. For comparison, we plot theprincipal and secondary spin wave dispersions obtained fromthe 1/S expansion. As we have already emphasized, the 1/Sexpansion underestimates the quantum renormalization ofthe exchange constants and, as a result, the agreement of thecalculated spin wave dispersions with the main peaks ob-served experimentally is poor.

The experimental energy and momentum resolutions havebeen accounted for to make contact with the experiment. Wefind that the effects of the finite energy resolution of �E=0.016 meV are generally outweighed by the effects of thefinite momentum resolution. This is a consequence of thelarge modulation of the spin wave dispersion along the chaindirection, i.e., �0k0�, �whose slopes can reach �E /�k�1.6 meV�, which causes an amplification of the effects ofthe momentum resolution. Given that the spin waves arenearly dispersionless along the �00l� direction, we have onlytaken into account the spatial resolution along the chain di-rection.

To illustrate this point, let us consider the results for scansB, E, G, and H shown in Fig. 10. The insets of panel �4�

FIG. 8. �Color online� Density plot of the scattering cross sec-tion as a function of energy and wave vector along the cut �0k 3

2�.

The light �dark� regions represent regions of small �large� scatteringintensity.

TABLE I. Parameterization of energy-momentum scans per-formed in Ref. 4: the momentum transfers k= �h ,k , l� are param-

etrized in terms of the energy transfer E �in meV�. k̂a is a measureof the polarization factor. Given that the weak interlayer coupling isneglected, h is not needed for the purposes of our calculation.

# k�rlu� l�rlu� k̂a

A −0.389+0.189E−0.016E2 0 0.05

B −0.30+0.189E−0.015E2 0 0.02

C 0.21+0.297E−0.026E2 0 0.25

D 2.11+0.29E−0.025E2 0 0.95

E −0.33+0.19E−0.015E2 0.78+0.37E−0.03E2 1

F −0.39+0.19E−0.02E2 1.66+0.37E−0.035E2 1

G 0.5 1.53−0.32E−0.1E2 1

H 0.28+0.29E−0.025E2 1.205 1

J 0.47 1.0−0.45E 1

K 0.29+0.29E−0.03E2 0.77−0.14E+0.013E2 1

FIG. 9. �Color online� The dispersion relation of magnetic exci-tations. The shaded regions labeled with capital letters A through Kindicate scan directions �the line thickness indicates the wave vectoraveraging�. The filled symbols are the main peaks in the line shapeas determined experimentally in the ordered phase �from Ref. 4.The dotted area indicates the extent of the scattering continuum.The open circles and squares are, respectively, the upper and lowerboundaries of the scattering continuum as determined experimen-tally. The upper thick dashed line is a guide to the eye. The thinsolid line is the experimental fit to the principal mode using effec-

tive parameters �J̃=0.61 meV, J̃�=0.107 meV�. The thick solid,dashed, and dash-dotted lines are, respectively, the 1/S results forthe principal ��k

0� and secondary ��k+ ,�k

−� modes determined fromEq. �40�.


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show the results of both linear spin wave theory and the 1/Sexpansion for a hypothetical energy resolution of �E=0.002 meV which has been introduced to make the variousdelta function peaks visible �the momentum resolution is setto zero �k=0�. First, we consider the results for scan H�panel �4� of Fig. 10�. Linear spin wave theory predicts peaksat approximately 0.27 meV and 0.37 meV corresponding tothe degenerate spin wave modes �+, �0, and �−, respec-tively. The 1/S correction yields a slight upward shift in theenergy of these peaks. In both linear spin wave and 1/Scalculations, the two-magnon scattering continuum is foundto carry nearly a quarter of the integrated spectral weight.Taking into account the finite momentum resolution �thewidth at half maximum is �k=0.057�, we find that the sharp

peaks get broadened very significantly as is shown in panel�4�. The dynamical structure factor now exhibits an extendedcontinuum in which the single-particle excitation can nolonger be resolved and merges smoothly with the two-magnon continuum. This result is qualitatively similar to theexperimental observations shown for comparison in panel �2�of Fig. 11.

Next we turn to scan G �panel �3� of Fig. 10�, whichprobes the vicinity of the wave vector �0,0.5,1.5�. Experi-mentally, a resolution-limited peak is observed at an energyof 0.107�10� meV in this region of intense scattering, seepanel �1� of Fig. 11. However, about two-thirds of the spec-tral weight is associated with a scattering continuum athigher energies. Both linear spin wave theory and the 1/S

FIG. 10. �Color online� Scattering cross section. The numbered panels �1�–�4� correspond to energy scans B, E, G, and H, respectively.The data has been convolved with the energy and spatial resolution. �E=0.016 meV for all plots and �k= �0.035,0.039,0.085,0.056� forplots �1�–�4�. The insets show the results of LSW theory and the 1/S expansion �LSW+1/S� for �k=0, �E=0.002 meV.

FIG. 11. Observed neutron scattering lineshape in scan G �1� and H �2� �data from Ref. 4 inthe ordered phase �T�0.1 K�. In scan G, theshaded area represents the the 1/S calculation�Eq. �39�� convolved with the experimentalresolution.


134429-10

expansion predict sharp peaks in this region of the Brillouinzone. The 1/S expansion gives a spin wave peak at �0

=0.18 meV carrying nearly half of the spectral weight andtwo further peaks at energies around 0.25 meV correspond-ing to the two secondary spin wave modes. The two-magnonscattering continuum extends up to 0.9 meV and carriesnearly a quarter of the spectral weight. We emphasize that, incontrast to �±, the principal mode �0 is close to a saddlepoint and, therefore, is nearly dispersionless. In panel �3�, thefinite energy and momentum resolutions are taken into ac-count. We see that the almost dispersionless principal moderemains sharp, but the secondary modes can no longer beresolved and are found to merge with the two-magnon con-tinuum. Irrespective of the discrepancies between the resultsof the 1/S expansions and the experimental data, our calcu-lation suggests that the lower boundary of of the measuredscattering continuum in scan G could be due to unresolvedtransverse magnons. Such a scenario had been previouslyconsidered and ruled out on the basis of the smallness of theratio Isec / Ipri of spectral weights of the secondary modes tothe principal mode predicted by linear spin wave theory.4

However, the results of the 1/S expansion show that spinwave interactions lead to an enhancement of this ratio for theG scan.

Next, we examine scan E �panel �2��, which probes wavevectors near k= �0,−0.25,1�. Linear spin wave theory pre-dicts coherent peaks at �0=0.35 meV for the principal modeand at �−=0.44 meV and �+=0.33 meV for the secondarymodes �see the inset in panel �2��. The two-magnon scatter-ing continuum is relatively weak and carries only about 23%of the total spectral weight. In the framework of the 1/Sexpansion, the principal mode is pushed upward in energy to�0=0.42 meV and occurs very close to the secondary mode�−=0.45 meV. The other secondary mode �+ is shifted verysignificantly to 0.39 meV, but carries only a minute fractionof the spectral weight. The two-magnon continuum is alsoshifted upward in energy and carries approximately a quarterof the total spectral weight. Once again the spin wave dis-persion is close to a saddle point and, as a result, the effectsof the finite momentum resolution are small. The main fea-ture in the structure factor is a broad peak formed by the twounresolved �− and �0 modes. This is quite similar to what is

observed experimentally �Fig. 5�e� of Ref. 4�. It is thentempting to speculate that the experimentally observed singlepeak is a result of the accidental near degeneracy of the �−

and �0 modes in the vicinity of k= �0,−0.25,1�. This wouldexplain both the absence of the �− peak in the experimentaldata and the anomalously large intensity of the observedpeak.

In panel �1� of Fig. 10, we plot the dynamical structurefactor for scan B near the wave vector �2,−0.25,0�. Here the

polarization factor �k̂a� in �39� leads to a strong suppressionof the out-of-plane fluctuations and the scattering is almostentirely due to the in-plane �± spin wave modes. The mag-non interactions renormalize �+ upward in energy to ap-proximately 0.42 meV, whereas the �− mode disappears inthe two-magnon scattering continuum. A careful analysisshows that the narrow peak at 0.55 meV is not due to asingle-particle excitation, but is a feature in the two-magnonscattering continuum.

The dominant contribution to the dynamical structure fac-tor in scan A in the vicinity of the wave vector �1.5,−0.3,0� comes from in-plane fluctuations because the polar-

ization factor k̂a suppresses out-of plane fluctuations. As canbe seen in Fig. 12, the magnon interactions lead to a spectralweight transfer to higher energies. The peaks near 0.8 meVand 0.85 meV can be traced back to single-particle poles inthe Green’s function. These poles are unphysical and are aresult of the uncontrolled nature of the 1/S expansion forsmall values of S. It is easily seen from the Dyson equation�21� that a large self-energy at a given wave vector can leadto “extra” poles in the Green’s function at high energiesabove the two-magnon continuum. The inclusion of higher-order terms in the 1/S expansion would provide decaymechanisms at all energies and lead to a broadening of thesehigh-energy peaks in the dynamical structure factor.

Last but not least, let us consider the vicinity of�0.8,0.4,0� �scan C�. As is shown in Fig. 12, the principalspin wave mode �0 is renormalized down to a slightly lowerenergy of approximately 0.42 meV. The �+ mode, whichoccurs at 0.35 meV in linear spin wave theory, disappearsentirely in the two-magnon continuum. The feature near 0.60meV can again be understood in terms of an enhancement ofthe two-magnon density of states. Compared with the neu-

FIG. 12. �Color online� Calculated scatteringcross sections in LSW theory and 1/S expansion�LSW+1/S� for scans A and C. Panels �2� and�4� show the LSW+1/S results with instrumentalresolutions of �E=0.016 meV and �k=0.02 �forscan A� and �k=0.04 �for scan C� taken into ac-count. See Fig. 5 from Ref. 4 to compare to ex-perimental data.


134429-11

tron scattering data �Fig. 5�c� of Ref. 4�, the structure factorshows features quite similar to the experimentally observedcontinuum. However, the scattering continuum occurs at en-ergies nearly 0.10 meV lower than what is observed experi-mentally.

VI. CONCLUSIONS

In this work, we have used nonlinear spin wave theory todetermine the dynamical structure factor in the ordered phaseof the spin-1

2 helimagnet Cs2CuCl4. We have taken into ac-count the first subleading contribution in a 1/S expansion,which incorporates interactions between magnons and gen-erates magnon decay processes as well as multiple magnonscattering continua. Both effects are particularly pronouncedin Cs2CuCl4 due to the noncollinear spin ordering, the lowspin value, and geometrical frustration.

We found that the results of nonlinear spin wave theoryexplain, on a qualitative level, many of the features observedin neutron scattering experiments. We find a strong scatteringcontinuum in the dynamical structure factor similar to theexperimental observations. Our calculations suggest the pos-sibility that some of the spectral weight at the low-energyboundary of the experimentally observed scattering con-tinuum in scan G could be due to single-particle excitationsthat are unresolved.

In the vicinity of saddle points of the spin wave dispersionrelation, the single-particle excitations are only weakly af-fected by the instrumental resolution and, hence, exhibitsharper peaks in the dynamical structure factor.

In spite of the qualitative agreement of the theory withexperiments, crucial discrepancies remain. First and fore-most, nonlinear spin wave theory fails to account for thelarge “quantum renormalization” of the main exchange pa-rameter. This indicates that �to order O�S0�� the 1/S expan-sion still underestimates the effects of quantum fluctuations.

Furthermore, there are significant quantitative differences be-tween our calculations and the experimentally observedstructure factor. One may speculate that a better agreementwith experiment could be achieved by taking higher-orderterms in the 1/S expansion into account.

The main lesson to be learned from our calculations isthat Cs2CuCl4 falls somewhere in between the two theoreti-cal scenarios that have been proposed previously. Our analy-sis shows that the physics of order plays an essential part inunderstanding the dynamic response Cs2CuCl4 at low tem-peratures: a large fraction of the spectral weight is carried byspin wave modes, which occur over a large range of frequen-cies. This is a strong indication that a putative spin-liquidground state is plainly not a good starting point for the de-scription of the ordered phase of Cs2CuCl4. On the otherhand, we have seen that �in low orders in 1/S� nonlinear spinwave theory significantly underestimates the effects of quan-tum fluctuations and, hence, expansions around the orderedstate also fail to account for the experimental observations.

Nonlinear spin wave theory can also be applied to inves-tigate the effects of magnetic fields. It is known that in thepresence of a field, linear spin wave theory is generally avery poor approximation as it excludes the all-importantmagnon decay processes.34 A self-consistent study of mag-netic field effects in Cs2CuCl4 is currently under way.35 Dur-ing completion of this work, we became aware of a paralleleffort which reaches similar conclusions.36

ACKNOWLEDGMENTS

This work was supported by the Engineering and PhysicalSciences Research Consul under Grant No. GR/R83712/01.We are grateful to John Chalker and Alan Tennant for valu-able discussions. Particular thanks are due to Radu Coldeafor numerous helpful discussions and suggestions as well asproviding us with Figs. 9 and 11.

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